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Tricycle Undercarriage Development

Tricycle Undercarriage Development January, 1941 AIRCRAFT ENGINEERING 13 Oscillation of Castoring Wheels A Simple Investigation of the Damping Required to Overcom e "Shimmy" in Tricycle Undercarriages J . Lockwood Taylor, D.SC. PROBLE M which has attracted some representing angle of rotation about the Separating real and imaginary parts, attention since the advent of the tricycle castoring axis, and tyre deflection respectively. undercarriage is tha t of the tendency to δ is related to the side load on the tyre, P, at oscillate or " shimmy " sometimes shown by the the point of contact with the ground, by means nose-wheel, which is normally free to rotate of the tyre elasticity, viz. about the vertical or castoring axis, subject to the restraint imposed by frictional or Before passing to consideration of the hydraulic dampers. Various attempts have dynamics of the motion there is a kinematic on substituting the value just obtained for K. been made to predict the amount of damping relation between the variables which helps Equating to the previous expression for b2/a2, required to prevent the building-up of a towards solving for the values of the unknown gives finally dangerous wobble ; in the simplified investiga­ constants ; this expresses the automatic track­ tion which follows it is believed that all the ing tendency of the wheel when displaced from essential variables have been retained, and the This value for the frequency does not enter its central position in the fore-and-aft plane : basic character of the motion correctly repro­ the point of tyre contact is displaced by the into the expression for the fluid damping duced. amoun t rθ due to pivoting (r being the amount coefficient K, but it is required in order to " Shimmy " is essentially a coupled oscilla­ of trail) plus a further δ due to tyre distortion, calculate the equivalent frictional damping. tion of the wheel in azimuth and of the point and the fact that the tyre rolls at an angle θ The maximum fluid damping torque, K.dθ/dt, of contact of the tyre with the ground per­ to the fore-and-aft line is expressed by the has the value Kka, and the corresponding pendicularly to the plane of the wheel, the equation friction torque, assumed constant, can readily be shown to be Kka x π/4. The value of a to latter involving sideways deflection of the be used in this expression is t o be obtained from tyre. The principal geometrical variables for in view of the obvious relation between 5, the a wheel of given size are the inclination of the th e known ratio b/a, in conjunction with the distance travelled, and the speed, v, and time, t. castoring axis to the vertical and the amount value of b, which is the maximum lateral de­ Performing the differentiation. of trail, defined as the distance of the tyre flection of the tyre. This may be chosen to contact behind the inter-section of the axis agree with the value of the side load on the and on equating real and imaginary parts in the with the ground. The castoring angle will in tyre which is just sufficient to produce skidding, usual way, the first instance be neglected, the axis being since no greater value is possible. assumed vertical, but it will be shown that the The castoring axis, which has so far been terms in the equations which depend on it are assumed vertical, may in practice be inclined whence unimportant. The angle has, however, an a t a small angle, e, to the vertical. The effect indirect influence, since it affects the moment of this is to add a term to the torque equation of inertia of the wheel unit mass about the whose value is Wε (rθ + δ) ; the correspond­ castoring axis, for a given amount of trail. ing additional terms in the complex equation To obtain the remaining equations necessary The static stability, with the wheel at rest, for K, k are (Wεra + Wεb.eia). Insertion of to evaluate the unknowns, consider the mo­ is also affected, but this may not be very numerical values shows that these are un­ ments acting about the pivoting axis : important in practice. important in comparison with the main terms (i) the inertia torque, I.d2θ/dt2, I being the of the equation, and no further account need In any coupled motion of the type under moment of inertia of the complete wheel unit, consideration, the different coordinates of the therefore be taken of them. including axle and fork, about the axis (ii) the motion have the same frequency and differ damping torque ; for the present this is as­ Other factors which have been ignored include only in amplitude and phase. Further, since sumed to be of fluid origin, and linear with the the damping of the tyre itself, the flexibility we wish to find the critical amount of damping angular velocity, viz. K.dθ/dt (iii) the moment of the fuselage structure supporting the nose- which will jus t suffice, on th e assumptions made, wheel, and gyroscopic couples. The latter of the side force at the tyre, Pr, and of the to prevent the amplitudes from increasing enter in only so far as the flexibility of the sup­ ground drag on the tyre, μWd, in terms of the over a period of time, it is legitimate to assume coefficient of ground friction, μ, and the vertical port permits lateral inclination, corresponding a simple-harmonic form of motion of steady load on the wheel, W. with precession of a gyro, and their effect is to amplitude, and work back to the calculation give additional damping. In all probability Th e equation expressing the balance of of the necessary damping. Accordingly we this also applies to the direct effect of the im­ moments is may put perfect rigidity of support, so that on all grounds the value obtained for the minimum Substituting for θ, Ρ and δ, and omitting the damping torque is a conservative one, which common factor eikt may safely be used. as the two fundamental dependent variables. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Aircraft Engineering and Aerospace Technology Emerald Publishing

Tricycle Undercarriage Development

Wernitz, W.
Aircraft Engineering and Aerospace Technology , Volume 13 (1): 7 – Jan 1, 1941
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Publisher
Emerald Publishing
Copyright
Copyright © Emerald Group Publishing Limited
ISSN
0002-2667
DOI
10.1108/eb030731
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Abstract

January, 1941 AIRCRAFT ENGINEERING 13 Oscillation of Castoring Wheels A Simple Investigation of the Damping Required to Overcom e "Shimmy" in Tricycle Undercarriages J . Lockwood Taylor, D.SC. PROBLE M which has attracted some representing angle of rotation about the Separating real and imaginary parts, attention since the advent of the tricycle castoring axis, and tyre deflection respectively. undercarriage is tha t of the tendency to δ is related to the side load on the tyre, P, at oscillate or " shimmy " sometimes shown by the the point of contact with the ground, by means nose-wheel, which is normally free to rotate of the tyre elasticity, viz. about the vertical or castoring axis, subject to the restraint imposed by frictional or Before passing to consideration of the hydraulic dampers. Various attempts have dynamics of the motion there is a kinematic on substituting the value just obtained for K. been made to predict the amount of damping relation between the variables which helps Equating to the previous expression for b2/a2, required to prevent the building-up of a towards solving for the values of the unknown gives finally dangerous wobble ; in the simplified investiga­ constants ; this expresses the automatic track­ tion which follows it is believed that all the ing tendency of the wheel when displaced from essential variables have been retained, and the This value for the frequency does not enter its central position in the fore-and-aft plane : basic character of the motion correctly repro­ the point of tyre contact is displaced by the into the expression for the fluid damping duced. amoun t rθ due to pivoting (r being the amount coefficient K, but it is required in order to " Shimmy " is essentially a coupled oscilla­ of trail) plus a further δ due to tyre distortion, calculate the equivalent frictional damping. tion of the wheel in azimuth and of the point and the fact that the tyre rolls at an angle θ The maximum fluid damping torque, K.dθ/dt, of contact of the tyre with the ground per­ to the fore-and-aft line is expressed by the has the value Kka, and the corresponding pendicularly to the plane of the wheel, the equation friction torque, assumed constant, can readily be shown to be Kka x π/4. The value of a to latter involving sideways deflection of the be used in this expression is t o be obtained from tyre. The principal geometrical variables for in view of the obvious relation between 5, the a wheel of given size are the inclination of the th e known ratio b/a, in conjunction with the distance travelled, and the speed, v, and time, t. castoring axis to the vertical and the amount value of b, which is the maximum lateral de­ Performing the differentiation. of trail, defined as the distance of the tyre flection of the tyre. This may be chosen to contact behind the inter-section of the axis agree with the value of the side load on the and on equating real and imaginary parts in the with the ground. The castoring angle will in tyre which is just sufficient to produce skidding, usual way, the first instance be neglected, the axis being since no greater value is possible. assumed vertical, but it will be shown that the The castoring axis, which has so far been terms in the equations which depend on it are assumed vertical, may in practice be inclined whence unimportant. The angle has, however, an a t a small angle, e, to the vertical. The effect indirect influence, since it affects the moment of this is to add a term to the torque equation of inertia of the wheel unit mass about the whose value is Wε (rθ + δ) ; the correspond­ castoring axis, for a given amount of trail. ing additional terms in the complex equation To obtain the remaining equations necessary The static stability, with the wheel at rest, for K, k are (Wεra + Wεb.eia). Insertion of to evaluate the unknowns, consider the mo­ is also affected, but this may not be very numerical values shows that these are un­ ments acting about the pivoting axis : important in practice. important in comparison with the main terms (i) the inertia torque, I.d2θ/dt2, I being the of the equation, and no further account need In any coupled motion of the type under moment of inertia of the complete wheel unit, consideration, the different coordinates of the therefore be taken of them. including axle and fork, about the axis (ii) the motion have the same frequency and differ damping torque ; for the present this is as­ Other factors which have been ignored include only in amplitude and phase. Further, since sumed to be of fluid origin, and linear with the the damping of the tyre itself, the flexibility we wish to find the critical amount of damping angular velocity, viz. K.dθ/dt (iii) the moment of the fuselage structure supporting the nose- which will jus t suffice, on th e assumptions made, wheel, and gyroscopic couples. The latter of the side force at the tyre, Pr, and of the to prevent the amplitudes from increasing enter in only so far as the flexibility of the sup­ ground drag on the tyre, μWd, in terms of the over a period of time, it is legitimate to assume coefficient of ground friction, μ, and the vertical port permits lateral inclination, corresponding a simple-harmonic form of motion of steady load on the wheel, W. with precession of a gyro, and their effect is to amplitude, and work back to the calculation give additional damping. In all probability Th e equation expressing the balance of of the necessary damping. Accordingly we this also applies to the direct effect of the im­ moments is may put perfect rigidity of support, so that on all grounds the value obtained for the minimum Substituting for θ, Ρ and δ, and omitting the damping torque is a conservative one, which common factor eikt may safely be used. as the two fundamental dependent variables.

Journal

Aircraft Engineering and Aerospace Technology – Emerald Publishing

Published: Jan 1, 1941

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